124 (1999)
MATHEMATICA BOHEMICA
No. 2–3, 167–172
SOME CLASSES OF INFINITELY DIFFERENTIABLE FUNCTIONS
G. S. Balashova, Moskva
(Received November 24, 1998)
Dedicated to Professor Alois Kufner on the occasion of his 65th birthday
n }
Abstract. For nonquasianalytical Carleman classes conditions on the sequences {M
n } such
and {Mn } are investigated which guarantee the existence of a function in CJ {M
that
u(n) (a) = bn , |bn | K n+1 Mn , n = 0, 1, . . . , a ∈ J.
n } and {Mn } are analysed. Some still unConditions of coincidence of the sequences {M
known classes of such sequences are pointed out and a construction of the required function
is suggested.
The connection of this classical problem with the problem of the existence of a function
with given trace at the boundary of the domain in a Sobolev space of infinite order is shown.
Keywords: Carleman class, Sobolev space
MSC 2000 : 26E10, 46E35
Nonquasianalytical Carleman classes of one real variable
(1)
n ,
n } ≡ {f (x) ∈ C ∞ (J) : max |f (n) (x)| K n+1 M
CJ {M
(f )
x∈J
n = 0, 1, . . .}
n } satisfies the following condiare considered. That means that the sequence {M
tions:
(2)
(3)
1/n = ∞,
lim M
n
n→∞
∞
nc
M
< ∞,
c
M
n=0
n+1
c } is the logarithmically convex regularization of {Mn } (cf. [1]).
where {M
n
167
Definition 1. The indices {ni } such that
Mni = Mnci
are called fundamental indices for the logarithmically convex regularization of {Mn }.
Definition 2. The sequence {Mn } is called almost logarithmically convex if for
all its fundamental indices the following condition is satisfied:
sup (ni+1 − ni ) = K < ∞.
i
If K = 1 then the sequence {Mn } is logarithmically convex.
The family of sequences {bn } such that
|bn | K n Mn ,
n = 0, 1 . . . ,
K = K({bn }),
is denoted as B{Mn }.
. Find conditions on the sequences {M
n } which guarantee
and {M
n } satisfying
for any sequence {bn } ∈ B{Mn } the existence of a function f (x) ∈ CÊ{M
the following conditions:
f (n) (0) = bn ,
(4)
n}
n = 0, 1 . . .
n for all n = 0, 1 . . .
It is clear that Mn M
n } and {Mn } are analysed.
In particular, the conditions of coincidence of {M
The problem was studied by T. Bang [2], E. Borel [3], T. Carleman [4], L. Carleson
[5], G. Wahde [6], B. S. Mitiagin [7], L. Ehrenpreis [8], G. S. Balashova [9] and other
authors.
Theorem 1. For any sequence {bn } ∈ B{Mn } and any number α > 1 there exists
n } satisfying the condition (4), where
the function f (x) ∈ CÊ{M
n = nαn
M
n
k=1
Mk
Mk+1
Mk
n−k
,
Mk =
Mk
,
k αk
k = 1, 2, . . .
.
We construct the desired function. It is known that there exists a
function ψ(x) ∈ C(∞
Ê) satisfying the following conditions:
1) ψ(x) 0, max ψ(x) = ψ(0) = 1, ψ (n) (0) = 0, n = 1, 2, . . .;
x∈R
168
2) ψ(x) = 0, if |x| > 2
3) max |ψ (n) (x)| and
|x|<δ
∞
n=1
n
j=1
∞
n=1
µ−1
n = δ;
µj , where µn > 0 is an increasing sequence such that µ0 = 1
µ−1
n < ∞.
The required function is
f (x) =
∞
bk
k=0
k!
ψk (dk x),
where ψk (x) satisfies the conditions 1)–3) with
α
µ(k)
n = (n + k) ,
dk = K(α)
Mk+1
.
Mk
. If the sequence {Mn } has the property that for some α > 1 the
n = Mn .
sequence {Mk k −αk } is almost logarithmically convex, then M
.
n = Mn . When
1◦ . If Mn = nαn lnβn n, α > 1, β 0, then M
β = 0, we obtain the known result of L. Carleson, L. Ehrenpreis and B. Mitiagin.
n = Mn .
2◦ . If Mn = anα (nβ lnγ n)n , a > 1, α > 1, β 0, γ 0, then M
If the sequence {Mn } grows slowlier than nαn , α > 1, then the following is true:
Theorem 2. If Mn = (n lnγr n lnβr+s n)n , γ > 0, β 0, r 1, s 1, then
n ) satisfying the condition (4), where M
n =
there exists a function f (x) ∈ CÊ(M
β
γ+1
(n lnr n lnr+s n)n (ln n ln ln n . . . lnr−1 n)n , lnr n means r-times iterated logarithm.
∞
k=0
is of a constructive character.
bk k
k! x ψk (dx),
The required function looks like f (x) =
(k)
where the constant d is chosen, and the sequence µn
is built as
(k)
(n + k) lnβr+s (n + k).
follows µn = (n + k) ln(n + k) ln ln(n + k) . . . lnr−1 (n + k) lnγ+1
r
.
n =
When r = 1, γ = 1, β = 0, Mn = (n ln n)n , we obtain M
(n ln n)n , which is the known result of L. Carleson.
2
While studying estimates of the norm of the n-th order derivative of a function
f (x) on the Lebesgue space of p-integrable functions (1 p < ∞) there was obtained
n } is logarithmically convex and for some α > 1
Theorem 3. If the sequence {M
−αn
} is almost logarithmically convex, then for any sequence
the sequence {Mn n
169
1
1− p , there exists an infinitely differentiable
1/p M
{bn } ∈ B{Mn }, where Mn = M
n+1 n
function on Ê such that
n ,
and ||f(x) ||Lp (Ê) K n+1 M
(n)
f (n) (0) = bn
n = 0, 1, . . .
. Theorem 3 makes sense only for such sequences {Mn }, for which
the ratio MMn+1
grows in n faster than the geometrical progression (for example,
n
ns
Mn = 2 , s > 2, n = 1, 2 . . .).
.
n+1 . That result gives the best
When p = 1 we have Mn = M
n+1 n as it is evident that M
n+1 Mn . In fact, K n+2 M
estimation for M
∞
∞
f (n+1) (x)L1 (Ê) |f (n+1) (x)| dx | f (n+1) (x) dx| = |f (n) (0)| = |bn |.
0
0
The problem of the existence of a function with the given trace at the boundary
of the domain G ∈ Ê in the space
(5)
W ∞ {an , p}(G) ≡
∞
u(x) ∈ C(G)
: (u) =
∞
n=0
an Dn u(x)pLp (G) < ∞
is very closely related to the one mentioned above (see [10], [11]). Here an 0,
1 p < ∞. These spaces are the energy spaces for the differential equations of
infinite order the model example of which is the following
(6)
∞
(−1)n Dn (an |Dn u|p−2 Dn u) = h(x),
x ∈ G = (0, a)
n=0
Dn u(0) = bn ,
(7)
Dn u(a) = cn ,
n = 0, 1, . . .
For the solvability of the problem (6), (7) we should first of all investigate the
conditions of existence of a function in the space (5), satisfying the conditions (7).
We will suppose that the space (5) is nontrivial which means that the space
◦
W ∞ {an , p}(0,a) ≡ {u(x) ∈ C0∞ (0, a), (u) < ∞}
contains at least one function other than that which is identical to zero. Yu. Dubinskij
[11] showed that this is the case if and only if the sequence {Mn } defined by Mn =
−1/p
an
for an = 0 and Mn = ∞ for an = 0, specifies a nonquasianalytic Carleman
class (1), i.e., the conditions (2), (3) hold for {Mn }.
170
Theorem 4. A necessary and sufficient condition for the sequence {bn } to be
extendable in any space W ∞ {an , p}(0,a) is
lim 1 |bn |1/n
n→∞ n
(8)
= K < ∞.
We shall call a trace satisfying the condition (8) analytical.
. For any space W
∞
{an , p}(0,a) there exists a nonanalytic trace extend-
able in this space.
Theorem 5. For the sequence {bn } to be extendable in the space W ∞ {an , p}(0,a) ,
the following condition is necessary:
∞
(9)
n=0
1/p
1
1− p
an+1 an
|bn |p < ∞.
Theorem 6. Let the sequence {an } be such that
(10)
1 > aqn an+1 ,
n = 0, 1, . . . ,
a0 > 0,
for some q > 1. Then for the existence of a function u(x) ∈ W ∞ {an , p}(0,a) with the
given trace {bn }, the condition
(11)
∞
1
c
|bn |p (Mnc )−(1− p ) (Mn+1
)
−1
p
<∞
n=0
is necessary and sufficient.
.
If the sequence {an } satisfies the condition (10) and the sequence
is almost logarithmically convex, then Mnc = a−1
and the condition (11)
n
coincides with the condition (9).
{a−1
n }
. Proofs of Theorems 4–6 can be found in the paper [10].
171
References
[1] Mandelbroit S.: Adjoining Series. Regularization of Sequences. Applications. Izdat. Inostrannoj Literatury, Moskva, 1995. (In Russian.)
[2] Bang T.: On quasi-analytiske funktioner. Thèse, Kyøbenhavn, 1946.
[3] Borel E.: Sur les fonctions d’une variable réelle indéfiniment dérivables. C. R. Acad. Sci.
174 (1922).
[4] Carleman T.: Les fonctions quasi-analytiques. Paris, 1926.
[5] Carleson L.: On universal moment problems. Math. Scand. 9 (1961), no. 2, 197–206.
[6] Wahde G.: Interpolation on non-quasi-analytic classes of infinitely differentiable functions. Math. Scand. 20 (1967), no. 1, 19–31.
[7] Mitiagin B.S.: On infinitely differentiable function with the values of its derivates given
at a point. Dokl. Akad. Nauk SSSR 138 (1961), 289–292.
[8] Ehrenpreis L.: The punctual and local images of quasi-analytic and non-quasi-analytic
classes. Institute for Advanced Study, Princeton, N. J., 1961. Mimeographed.
[9] Balashova G.S.: On extension of infinitely differentiable functions. Izv. Akad. Nauk
SSSR, Ser. Mat. 51 (1987), no. 6, 1292–1308. (In Russian.)
[10] Balashova G.S.: Conditions for the extension of a trace and an embedding for Banach
spaces of infinitely differentiable functions. Mat. Sb. 184 (1993), no. 1, 105–128. (In
Russian.)
[11] Dubinskij Yu. A.: Traces of functions from Sobolev spaces of infinite order and inhomogeneous problems for nonlinear equations. Mat. Sb. 106(148) (1978), no. 1, 66–84. (In
Russian.)
Author’s address: G. S. Balashova, Department of Mathematics, Power Engineering Institute, Krasnokazarmennaja 14, 111 250 Moscow, Russia, e-mail: balashov@cs.isa.ac.ru.
172